Optimal. Leaf size=35 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0081635, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {608, 31} \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(a+b x) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0065614, size = 26, normalized size = 0.74 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 0.7 \begin{align*}{\frac{ \left ( bx+a \right ) \ln \left ( bx+a \right ) }{b}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0589, size = 19, normalized size = 0.54 \begin{align*} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65035, size = 22, normalized size = 0.63 \begin{align*} \frac{\log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087553, size = 7, normalized size = 0.2 \begin{align*} \frac{\log{\left (a + b x \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15692, size = 23, normalized size = 0.66 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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